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AGGREGATE DEMAND WITH PARALLEL
MARKETS
ARINDAM DAS-GUPTA AND
SHOVAN RAY
No. 1/89 MARCH, 1989
17534
339 02BR lfl;l
This is a revised and extended version of an
earlier paper. Y/e wish to thank without implicating
Amaresh Bagchi, Partha Ganguli, Santiago Levy, Hiranya
Mukhopadhyay, Amal Sanyal, V.B. Tulasidhar and the
participants of workshops or seminars at the National
Institute of Public Finance and Policy, the Harvard
Institute of International Development, Jawaharlal
Nehru University and the 26th Conference of the Indian
Econometric Society for helpful comments on the earlier
paper. Thanks are due to Promila Bansal for expeditiou
word processing.
SUMMARY
The paper studies aggregate demand in an
economy in which households can evade taxes, accumulate
illegal wealth and in which there is unaccounted pro
duction. The objective is to evaluate tie effectiveness
of monetary and fiscal policies and also to evaluate
anti-black economy policies such as tax enforcement,
amnesties and demonetisation. Two regimes, a standard
IS—IM regime and a credit-constrained regime, are
studied, drawing on a model of household behaviour.
It is found that expansionary monetar3r policy gains
in effectiveness relative to exp an si on arir fiscal
policy as compared to the all-white economy in both
regimes though the former stimulates while the latter
curbs parallel production. Increased auditing to
detect tax evasion from white factor incomes stimulates
the black economy. Amnesties stimulate parallel activity
while possibly depressing the white economjr.
1, Introduction
The problem of large parallel markets coexisting
side by side in many economies has attracted much attention
in recent times, both in developed and developing countries.
Various policies have been advocated and used in attempts
to curb parallel activity. Policies include tax audits
and penalties, amnesties and demonetisation. Yet, to our
knowledge, no framework exists for the evaluation of the
effects of such policies on the parallel economy and on
macro activity. Even micro-theoretic evaluation is limited
to a subset of such policies, with attention focused on
tax audits and penalties. Equally, the efficacy of macro-
economic fiscal and monetary policies in the presence of
such markets has not been subjected to detailed examination.
To our knowledge, the paper by Sundaram and Pandit (1977)
represents the only attempt to study the effectiveness of
fiscal and monetary policies in the presence of a parallel
economy. They, however, do not consider the micro-founda-
tions of household behaviour and are therefore unable to
go much further than to point to the reduced effectiveness
of macro policies on aggregate demand. Their model is
also not equipped to evaluation anti-parallel economy
policies.
In this paper we propose a framework for studying
the demand side of such economies and the effectiveness
of the various policies referred to above. The paper may be
viewed as an attempt to develop the core of a complete
model in which aggregate supply is also incorporated.
This is currently absent from the model developed. Given
the objectives of the paper, we closely follow the standard
- 2 -
IS-IM framework. However, since in constructing the model
we have in mind the stylised realities of developing 2
coimtries , a short-run model of a credit-constrained
economy in the tradition of the financial repression
school started by McKinnon (1973) and Shaw (1973) is also
studied.
In section 2, a model of household behaviour in
the presence of tax evasion, unaccounted wealth accumu
lation and factor payments is developed. The purpose
of this exercise is to study saving behaviour and certain
other features needed for the development of the macro
framework. Section 3 develops the aggregate IS-IM frame
work for both flexible interest and credit-constrained
regimes. Section 4 contains a comparative static analysis
of the effects of fiscal and monetary policies. In
section 5, macro-economic effects of anti-parallel market
policies are examined. In section 6 we summarise the
main conclusions and indicate the direction in which
further work may be undertaken. Appendix I contains a
detailed discussion of the model of household behaviour
while Appendix II contains algebraic expressions for
stability conditions and policy rr Itipliers.
The structure of the proposed IS-IM macro model
differs from the standard IS-LM model in two important
respects. A portion of the total money supply, held as
black wealth balances, is unavailable to meet white money
demand. This affects the specification of the white IM
equation. In particular, since bla^ck saving (dissaving)
~ 3 -
leads to an increase (a decrease) in black wealth balances,
it enters the LEd equation. Secondly, three additional
equations are added, providing the link between black
and white interest rates and specifying parallel goods
and money market equilibrium conditions. Of course, IS
and LM equations need to be extended to incorporate the
effects of anti-black market policiss on household and
firms behaviour, a task facili u at e cl : oy the analysis of
household behaviour in the next section.
2. A Model of Household Behaviour with Parallel Markets^
The following analysis of household behaviour
is intended to consistently derive household saving
equations and government revenue equations for use in the
macro-economic model presented in the next section.
Households live for one period, starting life
with initial black and white bequests. They .receive
income in a large number of small parcels, each with an
identical and independent probability of detection if not
declared to tax authorities. There also exist firms which
indulge in white and black production. The government
undertakes income scrutiny of households and imposes
penalty on white factor income which is received but not
declared. All factor receipts from black production and
the part of factor income which is received from white
production but is neither declared not detected constitute
black income of households. Black income and wealth are
subject to wealth scrutiny. If detected, such funds are
confiscated.
- 4 -
Household lifetime consumption opportunity are
composed of the funds received (bequest and disposable
income) from white and black sectors, less the amount
they save. What they in fact consume depends not only on
the maximum opportunity they have, but also their preference
pattern, including in the latter the utility they receive
from the bequest they make. These utility considerations
in turn are influenced by a number of factors the household
may consider important, such as the rate of interest,
incomes received, etc. These considerations relating to
the micro foundations of household behaviour are
discussed in detail in A p p e n d i x I.
For the macro-economic analysis in this study,
we need to consider the aggregate fleakage* in the economy
for equilibrium in the goods market, following the
standard Keynesian method. Note that leakage consists
of household savings as well as collections realised b}>-
the government. While the details of the derivation of
the ’ leakage ‘function* are discussed in Appendix I, we
note here the basic equation to be used in the macro
models
L = L (r, Y, y; W, w, tf, P, V) / n
+ + + - - + ~ ~ ' ' '
In equation (1 ), the signs underneath the arguments of
the leakage .function refer to the corresponding partial
derivatives. The arguments in the function respectively
refer to the white interest rate (R), white income (Y),
black income (y), white wealth (r/)7 black wealth (w),
- 5 -
the penal rate of tax upon detection (tf), the probability
of escaping detection (P), the fraction of black income
and wealth remaining after detection and confiscation
(V) and amnesty options allowed by the government (A).
The penal rate of tax, tf, is composed of the tax rate
(t) and a multiplicative penalty factor (f > 1 ). The
assumption of a large number of income parcels with
independent probabilities of detection if not reported,
implies that ex-ante income equals ex-post income with
unit probability. Thus households report none of their
income to tax authorities since expected income is maximised
by evasion. Consequently only the penal tax rate affects
household behaviour. The normal tax rate does not enter
as a separate argument.
An additional behavioural equation playing a role
in the macro-economic model to be studied is the black
saving function. The manner in which this is generated is
discussed in detail in Appendix I. For the macro-economic
model, we represent it as
s = s(r, y; W, w, tf, P, V,A)+ + - - + + + “ K J
One assumption plays an important role in the derivation
of the illegal saving function. Given identical (post
tax and penalty) interest rates in both sectors, households
are assumed to prefer to save out of white income. They
save out of black income only if their total desired
saving exceeds their white disposable income. This
explains the apparently paradoxical positive effect of the
penal tax rate on illegal saving. Since a penal tax
rate increase only affects undeclared but detected
- 6 -
(and therefore white) income, households compensate for
the excessive fall in white saving by increasing black
saving. Other signs in ( 1 ) and (2) accord with a priori
expectations.
A final relationship obtained from the household
model is a link between the white and black interest
rates. The relationship is given by
r = XR, X = (i-P)(<*.tf)+VP < 1 (3)
The relationship may be explained as follows. XR is the
effective (after tax and penalty) white interest rate.
Since households are free to vary their saving out of
black and white income, they allocate their saving so
as to take advantage of the highest effective interest
rate. This will mean that no supply of saving will be
available In the sector with the lower effective interest
rate. Consequently effective interest rates will adjust
till they are equal.
3. A Model of Aggregate Demand with Parallel Markets
We now develop a model of aggregate demand by
extending the text book IS-IM model so as to incorporate
parallel production and tax evasion. The usual case of a
fixed money supply flexible interest rate regime is first
considered. Modifications are then made to permit the
study of a controlled interest rate regime as the latter
is of some interest to many developing countries.
- 7 -
(a) The flexible interest rate regime
(i) Preliminaries; The model to be constructed
has four asset markets, namely, the markets for black and
white money balances and black and white bonds. The
model also has two goods markets, one each for white and
black goods. In the usual all-white textbook model, one
asset market, the bond market., is dropped in view of the
stock budget constraint facing agents in the economy and
Walras* Law. In the current, parallel market model two
independent budget constraints, one each for white and
black asset balances have to be met. The independence is
a result of the assumption that assets recorded in official
books of account cannot, for the economy as a whole, be
exchanged for unrecorded assets and vice versa. That is,
no net laundering of black funds is assumed to take place.
In consequence, two asset markets, one black and one white,
need not be explicitly taken note of. The model has only
four independent market equilibrium conditions, two
each for money markets and goods markets.
The exclusion of net laundering does not preclude
gross laundering (or the swapping of black and white
asset balances between households) by appropriate book
entries. An example of such a transaction is the
following.
The popular method of dummy loans from friends
or relatives works as follows. A black wealth holder
persuades a relative to sign a document showing a loan
made to the wealth holder by the relative without any
money changing hands (except perhaps a payment for services).
The black wealth holder now uses his black wealth in
white markets. However, note that the relative’ s white
books must also show the loan given which forces her to
conceal an equivalent sum of cash. No net laundering has
taken place.
In fact, tl.e laundering rate, or the payment
made per unit of funds laundered by the black money
holder, is the price which clears black asset markets,
given the fact that black and white interest rates are
linearly related as has been argued in section 2.
Thus, in the model to be specified, goods markets
are cleared by the usual income adjustment mechanism, the
white money market is cleared by the white interest rate
and the black money market is cleared by the laundering
rat e.
(ii) Behaviour of firms; In contrast to the
household sector, firm behaviour is treated in a rudimentary
way. It is helpful to visualise production taking place
in two ’ rooms*, one containing capital goods whose purchase
is recorded in official books, which are financed by white
borrowing or equity, and the other containing unaccounted
capital goods financed by black household funds. White
production and black production can be correspondingly
distinguished. Unaccounted Taroauction facilities are
subject to detection and penalty. 1 Unaccounted investment
is undertaken to take advantage of the lower black interest
rate to the extent that this offsets expected penalties.
- g -
In accordance with this parable, firm behaviour is
assumed to be summarised by a pair of investment functions,
one each for black investment (i) and white investment (I ) .
The simplest forms, i(r ,V ), i < 0, iy >0 and l(R, tf, P),
1- < 0, I^£ < 0, Ip > 0, are assumed for these functions.
The term V is included in the black investment function
since increased wealth scrutiny decreases the value of
black factor payments relative to white factor payments.
Thus firms will have to pay higher wages relative to the
white sector which will reduce profitability and investment.
Similar considerations lead to the dependence of white
investment on the penal tax rate and the probability of
escaping audit.
Before leaving this section, one remark may be
worth making. Although it is normal to treat R as equal
to the return on government bonds in equilibrium, this
may no longer be true of evasion. It is reasonable to
suppose that interest payments on government bonds are
not as easily evaded as returns on private sector investment.
Thus, in equilibrium, the pre-tax return on investment
will be lower than the return on government bonds. Evasion,
therefore, may increase the cost of debt finance through bonds
in addition to lowering tax receipts.
(iii) The circular flow of income and equilibrium in
goods markets: The circular flow of income in
the model is shown in Figure 1. The flow diagram extends
the textbook diagram of a circular flow with producers,
households and government to incorporate black
income and expenditure. While the diagram is
- 10 -
largely self-explanatory, one point needs clarification.
Since firms and households maintain separate books of
accounts, an expenditure out of accounted funds with
households need not be shown as a receipt on the books
maintained by firms and vice versa. For example, white
consumption expenditure by households need not equal
sales receipts on consumption goods sold by firms and
shown on their books. Of course, total black and white
consumption expenditure (or investment expenditure) will
equal total receipts from sales of consumption goods
(or investment goods). There are three exceptions to this.
First, factor payments not accounted for on the books of
firms cannot be shown as income on household books.
Secondly, black investment must be financed entirely out
of black household saving since firms borrowing black
household funds will not be in a position to reveal
the source of these funds on their books and, conversely,
funds borrowed from white markets have to be accounted5
for in the books of firms. Finally, government expen
diture must clearly match firms' receipts from such
expenditure.
Black
an
d
Wh
ite
Facto
r
Pay
men
ts
Fig
ure
1
. C
ir
cu
la
r
Flo
w
of
Inco
me
fo
r an
Econom
y
with
E
va
sio
n
an
d
Para 1
1e
1 Product i
on
- 12 -
A consequence of this lack ox agreement is that
expenditure on white goods need not equal white production
and similarly for black production and expenditure.
However, such discrepancies will be random and, in
expected terns, production and expenditure will be equal
in both sectors. We may therefore still examine
expected levels of output. It will be understood that
expected or average levels of black and white output are
being studied below.
The upshot of the discussion above and in
section 2 is that the following goods market equilibrium
conditions can now be specified.
G+l(R)+i(r,V) = L (r ,Y ,y ;W ,w ,tf>P)V) (4)- - + + + + -- +- -
i(r,V) = (5)+ + + ““ “* + + + "*
Equation (4) is the economy-wide IS equation
for both black and white markets, while equation (5)
is the ’ is* equation for black goods market equilibrium.
G represents government expenditure.
( iv) Money market: Money demand by firms and
households has not been formally modelled since the
main interest in the household model is to derive the
leakage function, the black saving function and the
black-white interest rate link while investment demand
is the main focus of the discussion of firm behaviour.
- 13 -
Money demand functions of the traditional form are
assumed for white money demand (M) and black money
demand (m) so that
M = M (R, Y)- +
and
m = m ( r, y, Z)- +
In the black money demand function, Z is the laundering
rate. The sign of the partial derivative of m with
respect to Z is left undetermined as it plays no role in
the policy analysis to follow.
Fractional reserve banking is assumed away
so that white money supply consists entirely of the
monetary base, H, less leakage to the black economy.
A convenient way of representing this leakage is the
end-of-period black wealth. White money supply is
therefore given by H-Vw-s. Correspondingly, black
money supply must be Vw+s. The money market equilibrium
conditions are therefore
M (R, Y) = H-Vw-s (white , IM* equation) ( 6) “ +
and
Vw+s = m(r,y,Z) (Black fIM* equation' (7)— +
- 14 -
(v) The complete models The complete model is given by
the four equilibrium conditions (4), (5), (6) and (7) and
the interest rate relation (3 ). Substituting (3), Substi
tuting (3) into (4)y (5) and (6), the model may be seen to
consist of a three-equation sub-system jointly determining
equilibrium values of R, Y and y and an independent equation
(7)* which determines the equilibrium laundering rate.
Assuming stable adjustment to exogenous displacements, the
model can be conveniently graphed in a three-equadrsnt
diagram. This is done in Figure 2,
The economy-wide IS curve ( 4) is plotted along
with the white HA curve (6) in the first quadrant, which
has R on the vertical axis and Y on the horizontal axis.
The diagram is dram for a given value of black output, y,
A higher value of y would result in a left-ward shift of
the IS curve due to increased leakage from white money
supply. The intersection of the two curves gives the
equilibrium values of white income and the white interest
rate for the given level of black output*
The second quadrant graphs the relation r=XR from
equation ( 3 ) and determines the black interest rate given
the white interest rate. The third quadrant has the black
goods market equilibrium locus of r and y values. Starting
from any given equilibrium values of Y and R, equilibrium
values of r and y can be determined by following through the
dotted line EE in Figure 2 from the first to the third quadrant.
Clearly, the valuo of y in the third quadrant must bo consistent
with the value of y underlying the loci in the first quadrant
* KI
R
• n'urc•* ~S - *TV:t i U: •LM Model wi Lh Pa? 1
r+
- 16 -
for this to be an equilibrium. The working of the model
is explained more fully in the course of comparative
static exercises of the next two sections. Before these
exercises are undertaken, the controlled interest rate
regime is discussed.
(b) The Credit-Constrained Regime
In India and in certain other le^s developed
economies, formal sector interest rates are controlled
and credit is rationed. The reason for such a policy
is to provide cheap credit for socially productive
investment and also to make cheap debt finance available
to the government. The possibly adverse effects of such
policy have been the subject of discussion, in the
context of a growing economy, chiefly by the financialn
repression school of McKinnon (1973) and Shaw (1973)*
The main conclusion of this literature is that such
repression depresses saving and promotes overly capital-
intensive production in favoured industries. The
effectiveness of government policy in the presence of
such repression and the effect of anti-black policy in
this context are therefore of great interest.
The main alteration required to the model of
the previous section is to identify the effects of
constrained money supply on Investment and its rate of
return, given that the government bond rate is also fixed.
- 17 -
It is assumed, in the spirit of McKinnon (1973), that
firms require working capital in order to successfully
utilise investment goods they contemplate buying .
White investment is undertaken only if such finance is
granted by the money rationing authority. White invest
ment in this constrained situation is limited to
l(H-Vw-s,tf ,0 ), I>j >0,I^f < 0, I > 0, where it is
implicitly assumed that increased money supply on account
of a decrease in black saving is not mopped up by the
rationing authority. This is aversion of the celebrated
complementarity hypothesis of McKinnon (1973) according
to which financial and physical assets are complementarj7’
rather than substitutes in a supply-constrained8
situation. Given constrained investment,the effective
white return on investment will exceed the government bond
rate after taxes for most types of rationing policies
even though many investments which would not., -be under
taken if the government bond rate had been free to float
now find promoters. Furthermore, if thecredit constraint
is eased, both average and marginal rates of return on
investment should decline as more investment is undertaken.
Accordingly the equations determining income and the
white interest rate are now given *>y
l(H«Vw~s,tf,P)+i(r,V)+G = L
H = H (l), R* ' <0
(4*)
(6«)
The remaining equations of the first model, (3)»(5) and
(7), continue to hold.
- 18 -
An interesting property of the crcdit-constrained
regime is the one-way causality from the black market to
the white goods market. Given that interest rates are
now determined by the money supply and are unaffected
by changes in white output, chan, as in white output do
not have any impact on black output.
This completes the description of the
aggregate demand model under both regimes. We may now
turn to policy analysis.
4. Impact of Fiscal and Monetary Policies
a* Increased Government Expenditure
An increase in government expenditures (with
no change in the stock of money or the tax rate) causes
and increase in white income and interest rates but a
decrease in black income. The chain of causation runs
as follows. The increase in government expenditure
raises disposable income and aggregate consumption -
expenditure which initially stimulates white output.
In Figure 2 this would be represented by a rightward
shift of the aggregate IS curve. Hoy/ever, increased white i n c o
me raises u h i t e money dumund u h i c h p u t s u pw ar d p r e s s u r e on
u h i t e i n t e r e s t r a t e s . A s h j u s e h j l t i s ncu s t a r t uithdrauing s aw i n g
from the b l a c k s e c t o r , b l a c k interest rates a l s o increase.
Increased interest rates crowd out investment, dampening
the white income increase and reducing black income.
- 19 - '
While the rising black interest rate stimulates black
saving, the fall in black income reduces black saving by
an amount which more than offsets the increase induced
by the rising interest rate. The decrease in black saving
leads to reduced leakage from the white money supply
resulting in a rightward shift of the white IM curve
leading to a further stimulation of white output. After
secondary and tertiary effects the new equilibrium
configuration is as shown in Figure 3 ^y the line FF
passing through the intersection of the loci IS1 and IM1.
The initial equilibrium is shown by the line EE.
The effect on total output, Y+y, can be
found by adding up algebraic expressions for the
individual effects which are listed in Table A-4 of
Appendix II . The sign of the total effect is ambiguous.
A sufficient condition for increased government expansion
to have an overall expansionary effect is that the
sensitivity of black saving to unaccounted output (s )
exceeds the sensitivity of white money demand to white
output (My) • If this is the case, then white money
supply will increase enough to sufficiently dampen
crowding out. An interesting implication of this
result is that a trade-off between anti-parallel market
policy and the effectiveness of fiscal policy may exist.
Specifically, we have wealth scrutiny in mind. Wealth
scrutiny reduces disposable black income at any given
level of black output. Consequently, the sensitivity of
black saving to black output will be lowered at any
- 20 -
given level of black disposable income as wealth scrutiny
is stepped up. Therefore, in order to maintain the
overall expansionary influence of fiscal policy, lenient9
wealth scrutiny may be needed.
In the credit-constrained regime, the lack of
feed-back from the white goods market to the black
sector or to the white money market implies that govern
ment expenditure has no crowding out effect and, conse
quently, no effect on black income or any interest rate.
The multiplier for white (and total) income is the
inverse of the leakage propensity as in the simple
"Keynesian cross" model.
An increase in the tax rate has identical
effects to an increase in the fine rate and is taken up
in the next section.
b. Monetary Expansion
As can be expected, monetary expansion reduces
interest rates. ‘The immediate consequence of this is an
expansion in output in both black and white sectors.
However, if the interest sensitivity^ of illegal saving
is high enough, much of the additional money supply will
leak to the black sector. Both the fall in the interest
rate and the increase in black income stimulate black
consumption demand while having opposing effects on white
consumption demand. The upshot is that the initial
rightward shift of the IM curve is curbed and a leftward
- 21 -
shift of the IS curve results. The case where the final
III! curve coincides with the initial IM curve is shoun
in Figure 2. The line MM passes through the new equili
brium configuration.
The aggregate effect on (y+Y) must clearly be
positive (this is shown formally in Appendix I I ) . This
leads to the interesting conclusion that monetary expan
sion has increased effectiveness as compared to .fiscal
policy in an economy with parallel markets, though
expansionary fiscal policy still stimulates the white
sector.
The effects of monetary expansion in the credit-
constrained economy are qualitatively identical.. The
various cases are shewn in Table 1 .
5* Policies to Curbe the Black Sector
Here, five policies are considered. The three
* fiscal’ policies include an increased fine (or tax)
rate and increased income or wealth scrutiny. The two
’monetary1 policies are demonetisation and amnesties.
These are taken up in turn.
a. *Fiscal1 policies to curb black activity
All three fiscal policies lead to an increased
differential between white and black interest rates.
~ 22 -
TABLE 1
Effects of Fiscal and Monetary Expansion
Policy Flexible regime Constrained regjjne
Y R y (Y+y) Y R y (Y+y)
G- increase + + 9 + 0 0 +
II increase ? - + + ? + +
Effects
TABLE 2
; of Anti-Black Economy Policies.
Policy Flexible regime Constrained regime
Y R y (Y+y) •Y R y (Y+y)
tf increase ? ? ? 9 ? 0 9 ?
P decrease 9 ? + 9 ? 0 + 9
Demonetisation - ? ? - - - ...
Amnesties ? + + ? _ + +
- 23 -
In Figure 3, this is shorn by a steeper interest rate
ratio line, r=X^R, in the second quadrant. Consequently,
at a given white interest rate black investment is
stimulated. This stimulative effect conflicts with the
normal contractionary effect of increased taxes (or
fines) and results in ambiguous macro effects of a
change in the tax or fine rate. Similarly, conflicting
black saving and investment effects render the effect of
increased wealth scrutiny uncertain.
Stricter white income auditing (lower P) will
stimulate black production. This is because falling
interest rates stimulate consumption at any given level
of output, the stimulation being greater for black
consumption than for white consumption. Although invest
ment in both sectors is stimulated by falling interest
rates, white investment is curbed by stepped up income
auditing. Furthermore, unlike the case of raised taxes
and fines, the black saving rate at any given income
level is reduced on account of stricter auditing.
Consequently, the consumption effect is stronger in the
case of more severe auditing as compared to raised
taxes or fines. The result of this is increased black
income following stepped up auditing. This result may
be reinterpreted as follows. If white factor payments
become harder to conceal, economic activity shifts to
the black sector.
- 24 -
When all adjustments are taken into consideration,
other effects of stepped up auditing are uncertain* The
case wher.e white income decreases on account of a leftward
shift in the IS curve to IS1, with no net shift in the
IM curve, is shown in Figure 3- The new equilibrium
values lie on the line PP in Figure 4.
In the constrained regime, the effects on black
and white income are qualitatively similar. However,
there is no impact of increased taxes or income auditing
on the white interest rate. These results are summarised
in Table 2.
"k* Demonetisation and Amnesties
Demonetisation is taken to result in no exogenous
change in white money supply so that only black cash
balances are reduced. Consequently, the main effect
comes from reduced consumption on account of reduced
black wealth. The resulting leftward shift in the
IS curve puts downward pressure on interest rates which
stimulates black investment and curbs the consumption
effect. Consequently, while white income decreases,
the effect on black income is uncertain.
In the credit-constrained regime, interest rates
remain unaffected so that both white and black incomes
contract.
An amnesty is modelled as a one-shot reduction
in black wealth and an increase in white wealth. If
an amnesty is taken advantage of, it must be the case
- 25 -
R
Figure 5. Fiscal and t'tonetarv Policy Jiffects
wjth Parallel Production
- 26 -
R X ^ r R
Figure 4 . Impact of Stepned up Income Audi tins; arid AmnestyO < ■ ■■ - - — - — ~ i . ■ i— ii «i v ...............................■ -------- - — f -
- 27 -
that the reduction in after-scrutiny black wealth is
at least offset by the increase, in white wealth. Thus
if an amnesty is strictly preferred by an evader it must
result in lower initial tax revenues. An initially
revenue-neutral amnesty will not change aggregate
leakage but will stimulate white saving and black
consumption. The expansion in white money supply will
lead to declining interest rates which will further
stimulate black income generation* Since white money
expansion has uncertain effects on white income, black
and total income will rise while white income may or
may not. Thus, an amnesty has uncertain eventual effects
on government revenue but stimulates black activity.
Furthermore, it may not lead to an increase in white
income. In Figure 3, the new equilibrium with unchanged
white income is shown by the line'HA. The effects in
a credit-constrained regime are qualitatively similar.
Evidently, while demonetisation may be useful in
times of excess inflationary pressure, especially in a
credit-constrained regime, amnesty has much less to
commend it. The various results are summarised in
Table 2.
c. Which Policy?
To summarise, all anti-black economy policies
involve trade-offs at the least, in terms of white income;
- 28 -
Non^ is effective in curbing the black economy without
the possibility of imposing costs on the white sector.
Stepped up white income auditing and manesties, in fact,
lead to expansion of black activity. It is clear that
further work is required to identify useful anti-black
economy policies which do not impose costs on the white
sector and do not unintentionally stimulate black activity.
6. Conclusions and Extensions
The major findings of the policy analysis in the
two preceding sections can be summarised as follows.
In an economy with black production and tax
evasion, expansion in government spending has uncertain
effects on aggregate income while monetary expansion is
stimulative.. To restore the efficacy of fiscal expansion,
lenient wealth scrutiny may be necessary. However, with
respect to white income alone, fiscal expansion has its
expected impact while monetary effects are uncertain.
Finally, fiscal expansion curbs black production while
monetary expansion stimulates it.
With regard to anti-black economy policies, the
effects of increased taxes, fines or wealth auditing
are uncertain. While increased white income auditing
may curb evasion of taxes on white factor incomes, it
will stimulate underground income generation. Amnesties
stimulate black production and aggregate income generation.
- 29 -
Demonetisation curbs the white economy and will also
curb the black economy in a credit constrained regime.
Clearly, since amnesties do not curb tax evasion or
black income generation and may not stimulate government
revenues, they should be eschewed.
The framework, as it stands, is incomplete in
at least two major ways. First, laundering markets have
not been explicitly Incorporated even though the role
played by the equilibrium laundering rate has been
pointed out. Anecdotal evidence suggests that such
markets are crucial for businessmen engaged in illegal
production. Secondly, a more sophisticated model of firm
behaviour needs to be incorporated into the framework.
This extension should permit a better understanding of
investment behaviour and the supply side of the economy
in the presence of parallel markets. The incorporation
of a properly modelled supply side will also permit a
distinction to be made between real and nominal variables
and allow the study of inflationary implications of
parallel markets.
NOTES
For a recent survey see Cowell (1935).
Specifically, India.
Individual subscripts are supressed throughout the section except when aggregating over households in the final section. Throughout the paper additional subscripts denote partial derivatives with respect to subscript variables. Also, throughout the paper all variables are in real terns. A glossary of notation used is appended for easy reference.
For simplicity it is assumed that the cost of detection equals penalties collected, so that there is no net augmentation of government revenues. Also, for simplicity, it is assumed that there are no indirect business taxes.
This is true in aggregate. Of course, gross laundering transactions are possible.
•Local stability conditions and algebraic expressions for multipliers are given in Appendix II Also, a list of symbols used is given in Appendix III .
For a comprehensive survey of both theoretical and empirical work, see Fry (1988).
$here are some doubts about the validity of this
hypothesis based on cross-country evidence though the issue is not yet settled (Fry 1988),
We are indebted to Amal Sanval for helpful discussion on this point.
REFERENCES
Acharya, Shankar N ., et al. (1936), Aspects of the Black Economy in India,, National institute o’f fcbTic Finance' and' IPolicy, New Delhi.
Cowell, F .A .,(1985), Economics of Tax Evasion:A Survey," Discussion Paper No. 80,London School of Economics, London.
Fry, Maxwell J . , (1988), "Money, Interest and Banking in Economic development, Johns Hopkins ¥ress,'Baltimore.
Henry, James S., ( 19 8 3)> "Non-Compliance with U.S. Tax Law - Evidence on Size, Growth and Composition," in Income Tax Compliance, American Bar Association, ftesion, Va.'
McKinnon, Ronald I . , (2973)» Money and Capital in Economic Development. TiieBrookings Institution, Washington D.C.
Patnaik, Prabhat and Amal Sanyal, ( 1984), "Liquidity Prices and Growth: A Note," Economic and Political Weekly. Vol. 21, NoY 39, pp. 1705-1710.
Reinganum, J. and L.ti. Wilde, (1985), "Income TaxCompliance in A Principal Agent Framework," Journal of Public Economics, Vol. 29,PP*1-18
Sengupta, K . , (1987), "Tax Evasion Models: A Game Theoretic Approach," Indian Statistical Institute, New Delhi (mimeo).
Shaw, Edward S . ; (1973); Financial Deepening and Economic Development^, Oxford TjhiveVsity Press,^ New Dellii.^ ”
Sunderam?K. and V.N, Pandit, (1977), "Aggregate Demand under Conditions of Tax Evasion," Public Finance, Vol. 32, pp.333-342.
- 32 -APPENDIX I
A Model of Household Behaviour
1•1 Sources and Uses of Funds
We develop here a model of household behaviour over one
generation. A household begins with a bequest from its
progenitors and atrophies on making a bequest to its
descendants, Through its. lifetime the household receives
income as well as consumes out of the total resources at
its command - adding the flow of income to its endowment.
The household may also lend in the market for funds.
It may choose to participate in the black economy in all
its activities. A market is described as a fblack market1
when transactions in such a market are sought to be
unreported to the State, The rest is carried out in the
’white market*, Consider the sources and uses of funds
for such a household.
The household begins with an endowment which may
be white (W) or black (w) or both. It also receives
income from various activities undertaken. It declares
a part of its income and pajrs taxes on it; a part of its
undeclared income could also be detected, in the event
of which it has to pay penalties. These two together
constitute the household’ s white disposable income.
Total factor income receipts have two components. That
part which is reported in firms’ books of account is
denoted by Y, and that which is not, is denoted by y;
the second source clearly constitutes a part of black
income. In addition, black income arises due to the
undeclared and undetected part of Y.
- 33 -
Black income and wealth, have the additional risk
being subjected to scrutiny over time. V represents
the fraction of black funds remaining after "wealth
scrutiny" and P denotes the (constant) probability of
escaping detection. Reinganum and Wilde (1985) and
Sengupta (1987) show that, when auditing is costly, an
audit cut-off policy is superior to the random audit
policy implicitly being assumed here. However, they
implicitly assume that the probability of audit is equal
to the probability of detection and conviction. When
this is not the case, .the optimal P may not in fact be
achievable. In such an event no household would report
at the audit cut-off so that random "auditing'1 and a
cut-off policy become equivalent.
The household is able to consume in the
proportions it chooses out of white and black income.
Consumption out of white ana black income are denoted by
C and c respectively. White disposable income that is
not spent on consumption is white savings (S), which earn
pre-tax interest at the rate R and effective rate
(after tax and evasion) of R . Correspondingly, black
disposable income that is not consumed is: saved(s),
earning interest at the rate r. Households may dissave
if they so choose with the constraint that the present
value of loans contracted in the black market does not
exceed expected undetected interest income. This is a
solvency condition imposed on the household's borrowing
capacity. Savings and interest earnings are bequeathed
as B or b (white or black).
- 34 -
We assume a proportional tax rate on declared
income denoted by t. In the event that undeclared
income is detected, the penal rate of tax the household
pays is ft, f > 1. Total government revenues from all
sources are denoted by T.
Household income is assumed to be received in
small identical pay packets from several sources.
Undeclared income from each pay packet is assumed to have
an independent chance of being detected. Thus ex-ante
expected (after tax and fine) white and black income
can be treated as equal to ex-post income with unit
probability. This follows from a straightforward
application of the binomial probability law given the
constant t, f, P and V assumed. Therefore, given
undeclared income Yf we have the following expressions
for white and black disposable incomes
Yd=(Y~Y' )(l-t) + Y'(l-P)(l--tf) (A1»)
and
y £=VPY* ~ (1 -V) w + vy (A21)
In (A2f ), (1~V)U is the outflow of black wealth due to
additional wealth scrutiny operatio.ns. The parameter
V is determined in a fashion similar to the effective
tax rate. Black income inflows and black wealth are
held in several parcels each with an Independent proba
bility of detection (1-V). If detected, a fraction f 1,
- 35 -
assumed equal to unity for simplicity* is confiscated.
Expected after-scrutiny black funds are, therefore,
VCw+PY’ +y). As before, expected and ex-post after
scrutiny black funds are taken to be identical.
Households are assumed to maximise the utility
function
iKCiO+U* (B*) where C*=C+c and B*=B+b
Given the equality of ex-post and expected values,
utility from constant P ,f ,V and t, a household will
either report all its income or none of its income.
No income reporting will be strictly preferred if and
only if
(1-P) (1-tf) + V P > ( 1-jg) (A3)
This condition is assumed to hold throughout the paper.
We thus conclude that Y=Y* . Consequently, white and
black disposable incomes are redefined as
Yd=Y(l~P) (1-tf) (A1)
yd=VTY - (1—V)w + Vy (A2)
Furthermore, the effective after-tax white interest
rate is given under these assumptions, by
- 36 -
Rg = R (1 —P) (1-tf) + VRP (A4)
The household white and black budget constraints are
(W+Y(1™P)(1 -tf)-C) ( 1+R(1-P)(l-tf)) = B (A5)
•and
(V(v/+y+PY)-c)(l+r)+VRP(Y/+Y(l-i>)(l-tf)-C) = b (A6)
(A5) and (A6) can be coabined to yield the total budget
constraint
(W+Y(l-P)(l-tf)-C)(l+Re)+(V(w+y-fcPY)-c)(l+r) = B* (A71 )
Note that maximum black consumption is given by
c* = V(w+y+PY)+VRP(W+Y(1-P)( 1-tf)/(l+r) (AS)
when both white consumption and black bequests are zero.
The term VRP(W+Y(1-P)(1-tf)/(l+r) represents black
borrowing capacity. It should be noted that households
actually leave pre-tax bequests (S+V/ and s+Vw) for their
descendants. However, they derive utility from post-tax
bequests, B and b, so that these have to be determined.
Since the number of scrutiny operations can be
expected to be an increasing function of time, black
funds held for two periods will be less in expected
terms than black funds held for one period after penalty
- 37 -
payment. This could be formally accommodated by replacing
C. in (6) and (7) by Vf G^, where 1> V1 >V. This will
complicate the algebra and add nothing to insight and
is, therefore, eschewed. Instead, we assume that
household1s consumption expenses are met first out of
black funds and borrowing. They call on their white
funds to meet consumption expenses only in the event that
other sources are exhausted. The obverse of this behaviour
is that, to the extent possible, savings are made from
white funds. With this assumption we are now in a
position to explore the implications of the model.
1•2 The Black Loan Market
If Re >r9 it will be optimal for all households
to borrow on the black loans market and carry out all
saving in the white market. No household will be willing
to lend on the black market. The opposite is true if
R < r. Consequently, in equilibrium Re=r which implies
that r < R. This discussion ignores the probability of
residual saving in the sector with the lower interest
rate due to budget constraints.
1 • 3 Consumption and Saving Functions
Given the argument of the previous subsection
we may rewrite(A7') as
(N.-C*) ( 1+r) = B* (A7)
- 38 -
where N^=W+Y(l-P)(l-tf)+V(w+y+PY) is the beginning of
period household net worth. Under standard assumptions,
(A7) and the utility function imply a household consump
tion demand function of the fom C*=C*(R ,N1)=C*(r,N )6 | \with C* K 0 and (}*> 0. The normal restriction on the
r Nmarginal propensity to consume, 0 < < 1 is assumed*
Total savings are given by Y-C*~T=S* and total ‘ leakages*
are given by Y-C*=T+S* = L, where current period
government tax collection from the household is
T = Y--Yd~yd = (1-(1—P)(l-tf)~VP)Y + (1-V) (w+y) (A9)
Straightforward calculation yields
S* = S«(r,Y ,y ;W ,w ,tf,P) (A10)+ + + — “■*— +
and
L = L (r ,Y ,y ;W ,w ,tffP) (A11)+ + + — — + —
Fomulae for partial derivatives of (A10), .
(A11) and the consumption function are given in Table A-1.
The variable A is an artificial variable which captures
the direct effect of an amnestjr on the household. An
amnesty may be viewed as a one-shot opportunity allowed
by the government for households to declare concealed
wealth and .jive up a fraction to the government., (l~a)
is the fraction of declared wealth taken by the govern
ment. Thus the household gives up black wealth Vw in
exchange for white wealth aW. We assume that the amnesty
is revenue-neutral so that a=V.
- 39 -
We now turn to the study of black savings. To
do this, we take into account the possibility of house
holds being in one of three possible zones. These are
illustrated in Figure A1 where Ng* end of period net
worth, is plotted against consumption.
These zones may be described as follows s
High savers; they save all their white funds
and a part of their black funds.
The intermediate case; they consume all their
black funds making partial use of their
borrowing capacity in the black market and
save all white funds.
High consumers; they consume all their black
funds and in doing so exhaust all opportunities
of borrowing in the black market and also
consume a part of their white funds.
Households in zone II I , whose consumption is
large relative to total black income and wealth, may be
thought of as salary earners. In practice, such house
holds are likely to have both limited black wealth and
limited evasion opportunities, though differential
evasion opportunities are not modelled explicitly in
the formal analysis. These households not only save
out of white funds but consume almost entirely out of
Zone Is
Zone IIs
Zone Ills
40 -
BLACK WHITE
F ig u re A 1. Types o f t-fous.e ho I cls
- 41 -
the same funds as well. It is therefore reasonable to
hold that their importance to the parallel economy,
even when their numbers are large, is likely to be small.
For US evidence of this, see Henry (1983) and for Indian
evidence see Acharya et al, (19&5). Therefore, wherever
exogenous effects for zone III households conflict with
those for zone I and zone II households, the former
effect is assumed to be swamped by those of the latter.
This assumption needs to be invoked only in the case of
the interest rate effect on black saving. The dependence
of black saving demand on different exogenous changes
for the three household types is given in Table A2.
Finally, since increasedY causes both Y^ and y^ to
increase, the effect on black saving is ambiguous for
households in all zones. The effect will be small in
aggregate and is, therefore, neglected.
These assumptions imply an aggregate black
saving function given by
s = s(r, y, W, w, tf, P, V, A) (A12)+ + *"*'■“ + + + —*
Two remarks may be made before leaving this
section.
(a) The weak connection between white consumption
and white disposable income, which is a consequence of
time-dependent effects of wealth scrutiny, appears at
first sight to conflict with national accounts figures
for most coimtries. The paradox gets resolved once
- 42 -
it is recognised that consumption expenditure on the
white market (from all sources) is what is measured and
not white consumption expenditure*
(b) The high marginal propensity to consume
out of black income is often cited as one of the dama -
ging effects of parallel economies. Our model shows
that these are accompanied by offsetting effects in the
white propensity to consume so that the aggregate propen
sity is relatively unchanged. Thus, this criticism may
be misplaced.
~ 43 -
U
B
Y
y
uf
p
r
V
Note
TABLE A-1
E f f e c t s of E x o g e n o u s C h a n g e s on H o u s e h o l d C o n s u m p t i o n
and S a vi ng
C* S *
C * N
VC*
xc*
N
N
VC*
-YC*N(1-P)
~YC*|\.(l-uf-\/)
-C*N
V/CN
x (i- c*N)
U(1-C*N)
-(1-P)(Y-YC* .)
‘ C*N
"UC*M
1-XC*N
1-UC*N.
YC*N(1-P)
-(Y-YC*m) ( 1-uf-V) YC*W( 1 -uf-W)
c*r
( B+y+PY) C*
-c- -C*
( B + y + P Y ) ( 1 -C * ) - (B+ y+ pY ) C *
i x = ( 1 -p) ( 1 -uf) + \yp
- 44 ~
TM3LE A -2
E f f e c t s of E x o g e n o u s C h a n g e s on B l a c k S a v i n g
E x o g e n o u s H o u s e h o l d zone
\j a r iab le
“ J' " ‘ " i f ' ....i i f *
u
y ? ? ? ? ( s )
y + + 0 +
t f + + + +
P + + + +
R + + ?(+)
1/ + + -r +
A - - 0
No te s ? 1. S i Second o r d e r e f f e c t .
2 . H o u s e h o l d s c a n n o t move out o f t h e i r z o n e ,
by a s s u m p t i o n .
- 45 -
APPENDIX II
Comparative Static Algebra for the Macro Models
Preliminaries: In order to be able to write down
algebraic expressions compactly, we define some additional
notation in Table A-3.
Alphabetic subscripts denote partial derivatives
with respect to subscripted variables unless otherwise
defined in Table A-3- In the case of the penal tax
rate, tf, the subscript t is used.
The flexible interest rate regime; Differentiating
equations ( 4), ( 5 ) and ( 6) after substituting (3 ), one
gets the total differential system for policy analysis*
A necessary condition for local stability is that the
system determinant is negative. The system determinant
is given by ~n in Table A-3. Assuming local stability,J
comparative static multipliers will have the general form
-(numerator)/n . where the numerators are given inu
Table A-4. The signs of numerators can be directly
determined by noting the signs of partial derivatives,
as in Table 1 , except for three cases. These are as
follows:
1 . A sufficient condition for d(Y+y)/dH to be
positive is Ly>D . But I^-L =C* -C*y. From
Appendix I this can be found to be (V-X)C*^=
(1-P)(V+tf-1 )C*N* last expression is
positive whenever evasion is profitable.
- 46 -
TABLE A-3
A d d i t i o n a l N o t a t i o n f o r A p p e n d i x 2
■= Mp+Xir < n Kj = Lj s y~s j Ly5 = r * u
E 2 = IR + X ( i r -Lr ) < 0 K. = (Lr I . ) s y-s.Ly , jas p, t
£3 = ip ^ j — M yX F j —( L yE j-f I pH y) sj , j = t , p , y
E 4 = i r ” ^ r <<0 YX = " ' ^ r ^ S+fV1R Kr^
E 5 = n R ^ Ly ~ sy) “ ^ y RX = IY,YF y “ L Ys y1 r
Fj = Lj E:3 ~sj E:4 ? J = Y * u VX = E3 L y ^ R ^ YJr ) > 0
Fj = ( E j —I j ) E^ -Sj »j = t * p Y|_j = Ip Sy -X Fy
D = s ( h X E g - l ) ( c r e d i t ■ c o n s t r a i n e d re gi me o n l y )
h = -RI I 1/ ( l - X R i I 1 s r ) > 0 ( c r e d i t - c o n s t r a i n e d r a g i m e o n l y )
- 47 -
TABLE A-4
Numerators of Comparative Static Multipliers for the Flexible Interest Rats Regime
Exogenouschange
G increase
H increase
tf increase
P decrease
Amne sty
syEi
- V t +Yxxt
e i W p
Comparative static effect on
R y ^ _
yXE 2
L y X E 3
V yxxt
-np-yx Xp
Demonetisation E K1 u
"W ys y
Lysy
" y W t
'W YKP " RXXP
-m v kY u
-e-(LySfl+(l-V) Yh -Cl Y*-Ys fl+
( l - U ) L y S y
-nu
-SA *"Y 1+^ Y ^
+ (l-tf)LYXE3
- 48 -
2. In comparative static expressions for the case
of demonetisation, the term appears. The
sign of K may be found “by using the householdw
model of Appendix I. From there it can be found
that 3yi-VC*N, Iy=VC*N, sw=--( 1 -V)-cw and sy=V-cm.
Now VC ,T < c since the two are equal for house-i\ wholds in zones I and II and the former is strictly
smaller for households in zone I I I . Making
these substitutions we get Kw=VC 1~V) < 0.
The multipliers dY/dw and dR/dw may now be seen
to be negative as claimed,
3. As discussed in the text, demonetisation is
assumed not to lead to any exogenous change in
white money supply, while amnesties initially
increase white money supply by (l-V)dA.
The Credit-constrained regime; Equations (5) and (6*)
form an independent subsystem of the model determining
R and y. To solve the model, first differentiate (61)
totally and simplify to get
dft* - hd(H-Vw)+hs dy+h(s .dj)7Cu (A13)t7 J
where (s.dj) is a row vector of differential policy
effects on illegal saving and investment, j=tf,P ;w,A,
and u is a column vector of ones.
- 49 -
Next, totally differentiate (5) and use the resulting
total differential equation with (A13) get varioHS
policy multipliers for R and y. These are given in the
second and third columns of Table A-5. Using the results
of these two columns and the equations R^=R(l), i=i(XR),
the following equations can be deduced.
I=I(K, tf, P, A) (A14)
i=i(H, tf, P, A) (A15)+ + ~ +
y=y(H,tf,P,w,A) (A16)+ ? ~ + +
Substituting (A H ) - (A16) into (41) and totally
differentiating, allows us to solve for the impact of
policy changes on Y. These are given in the first
column of Table A-5. In deteraiining the sign of
d(Y+y), the term (Ly-L ) appears in the multipliersA v
for H and A. As discussed above, these are positive.
Table 2 gives the signs of these multipliers.
TABLE A -5
C o m p a r a t i v e S t a t i c M u l t i p l i e r s f o r the
C r e d i t - C o n s t r a i n e d R eg ime
E x o g e n o u sChange
isratiye^ s t a t i c e f f e c t on
r y
2 i n c r e a s e
H i n c r e a s e
tf i n c r e a s e
P d e c r e a s e
D e m o n e t i s a t i o n
1/L
( l ^ + i ^ “ L y y ^ ) / L y h s y/ D h X ^ / D
<i t+ I t-Lt-Lyyt) / L Y °
( -i p+ L p-I p+- L y p ) / L y 0
( L + L y ) / Lu y u Y
( s ^ s X )/ t r t
S n + SPTOrAPX n ) /
Sl / S y
Amne sty ( l fi+i ft-Lyyfl) / L Y n ( l - U ) Sy/ D h ( l - U ) X E 3/ D
APPENDIX III
Table of Symbols Used
The paper contains a large number of algebraic
symbols, of these symbols are used only .in a
single paragraph and are, therefore,, omitted from this
table without inconveniencing the reader. Other symbols
are as follows:
1* Used in the model of household behaviour
A
B,b
C,c
*c
f
L
H1tN2
R
Eer
S, s
T
t
UfU*
Variable measuring the effect of amnesties,
White and black bequests.
Household white and black consumption;
C = O c .
Upper limit of household black consumption.
Pine rate on (detected) unreported income,
Leakages = S+T,
Beginning and end of period household net
worth respectively.
Probability of escaping detection of unreported
income.
Pre-tax white interest rate.
Effective post-tax white interest rate.
Black interest rate.
Household white saving and black saving^S =S+s,
Tax revenues.
Income tax rate.
Utility indices.
- 52 -
V i Fraction of black wealth retained by the
household after appropriation by the govern
ment due to wealth scrutiny.
W,w o Household white wealth and black wealth
respectively.
X % Ratio of the black interest rate to the white
interest rate.
Y,y % Gross household beginning of period legal and
black factor incomes.
Y* ; Undeclared legal income.
Yd*2^ 1 White and black disposable income.
2• Additional symbols, used in the model of aggregate d emend
G 5 Government expenditure.
H
I . i
Z
High powered money stock.
White and black investment expenditure.
Fraction of white funds realised per unit of
black funds launc.ored.
Notes Subscripts denote partial derivatives with respect to subscripted variables except in the case of Y-,,Yn and R .a e